It depends somewhat on what medication was used for anesthetization, but most of them produce a brain state that is distinct from sleep. I won't comment on whether a person will subjectively experience a feeling of being rested (even though most people don't report feeling rested), but the important thing is again that being anesthetized is not the same as sleeping.

But the person I responded to didn't say "chemically induced coma", they just said "coma".

If we're talking about chemical coma/anesthetization, I'd say it would be hard to determine if a person was tired because they didn't "sleep" for the 5 hour operation, or because their body was traumatized during said operation.

As Brain_Doc82 said, the states of sleep and anesthesia are physiologically different, so we wouldn't necessarily expect one to be able to exactly substitute for the other. However there are many interactions. Anesthesia affects many of the systems that regulate sleep, and many anesthetics actually achieve their actions by targeting those systems.

Sleep need is generally understood in terms of what is called sleep homeostasis. The longer you are awake, the more your sleep homeostatic pressure increases, and the longer you are asleep, the more it decreases. We now know that the sleep homeostat probably represents the accumulation and clearance of certain sleep regulatory molecules in the brain (http://www.nature.com/nrn/journal/v9/n12/full/nrn2521.html).

We still don't know exactly how anesthesia affects the sleep homeostat, but there have been some studies. This 2004 review says:

Work in rats suggests that during prolonged anesthesia, sleep need does not accrue, and that recovery from sleep deprivation can occur during anesthesia. Moreover, both circadian rhythmicity and sleep deprivation clearly alter anesthetic action, indicating a link between factors known to regulate sleep and anesthesia. Little is known about how the anesthetized state might modulate the mechanisms governing sleep homeostasis. Anesthetics may act either directly, by reversing the buildup of sleep ‘debt’ in the brain, or indirectly by preventing normal wake-associated processes from doing the same.

However, all of these studies have so far been restricted to rats, and they haven't looked at the effects on waking cognitive function, only the effects on sleep need and other sleep markers. I also don't know of any studies to look at the effects of anesthetics on sleep homeostasis in humans.

Sleep is as much a neurological state as a physical one. The brain is, in fact, highly active during sleep in ways that measurable differ from wakefulness and comas.

Comas dampen brain activity, essentially "shutting down" parts of the brain. Medical comas also reduce bloodflow (and lower intracranial pressure), which is why they are important for things like complex neurosurgery.

Soporifics/hypnotics rarely result in restfulness, but are used to help force a natural sleep state (e.g.: to induce rest longer than the effects of the drug last).

It is unlikely that being "put under" for8 hours will have the same biophysical effect as sleep.